I am defending my PhD in just a little more than a week. I have been reflecting on this part of the math-doing processes a lot lately, especially as it pertains to teaching. I’ve been interested in experimenting with a “modern” teaching of mathematics (referencing the modernist movement in art and poetry). See, for example, the poem “Portrait of a Lady” by William Carlos Williams. The author here attempts to leave the work that goes into trying to describe the lady on the page, rather than giving us a finished, wrought piece of poetry.
I am currently toying with an idea. Next semester, when I teach Linear Algebra, I am thinking about assigning a proof that I don’t expect many students to complete, but as part of the assignment having them turn in their attempts, the ideas that didn’t work. I want to encourage students to try things that might be wrong rather than being worried about getting it perfect on the first try. I’d be interested to hear your thoughts!